Răspuns:
1. log₇(7x+9)=2
DDVA
7x+9>0
7x>-9
x>-9/7
x=(-9/7,+infinit)=dva
rezolvi ecuatia
1. 7x+9=7²
7x=49-9
7x=40
x=40/7> -9/7 apartine DVA
2. log₀₅(x-2)>log₀₅9/(x-3)
dva
x-2>0 x>2
9/(x-3)>0
x-3>0 x>3
DVA=(3,+infinit)
rezolvi inecuatia
x-2<9/(x-3)
(x-2)(x-3)<9
x²-2x-3x+6<9
x²-5x-3<0
x²-5x-3=0
x1==(5-[tex]\sqrt{37}[/tex])/2<3 nu apartine DVA
x2=(5+√37)/2>3 solutie
x apartine (5+rad 37/2,+oo)
3. log₍ₓ₊₂₎lx-9l≥1
DVA
x+2>0 x> -2
x+2≠1 x≠-1
x-9≠0 x≠9
DVA=(-2,1)U(1,9)I(9,∞)
rezolvare
x∈(-2,-1)
x+2∈(0,1)
l x-9l≤1
-1≤x-9≤1
x∈(8,10)∩(--2 ,-1)=Ф Nu admite solutii
Caz 2
x∈(-1,1)U(1.9)U(9,+∞)
x+2>1
x-9≥x+2 imposibil
inecuatia nu admite solutii
ex 4
log₁/₂(3x+5)≥ -2
DVA
3x+5>0
3x> -5
x>-5/3
x∈(-5/3,+∞)
rezolvare
3x+5)≤1/2⁻²
3x+5≤1/4
3x≤1/4-5
3x≤-19/20
intersectezi cu dva
(-∞. -19/20)∩(-5/3+∞)=
(-5/3, -19/20)
7
DVA x∈R
(2√2-√7)(2√2+√7)=8-7=1
(2√2+√7)=1/(2√2-√7)
(2√2+√7)ˣ=y>0
(2√2-√7)ˣ=1/y
Ecuatia devine
y+1/y=4√2
y²-4√2y+1=0
Δ=4√2)²-4=28
y1=(4-√28)2=(4-2√7)/2=2-√7<0 nu se accepta
y2=2+√7.>0 solutie=>
(2√2+√7)ˣ=(2+√7
x=log(2+√7)
Explicație pas cu pas:
